Calculus Examples

$x^{2} - 3^{x} + \frac{1}{x}$

Step 1

Write $x^{2} - 3^{x} + \frac{1}{x}$ as a function.

$f (x) = x^{2} - 3^{x} + \frac{1}{x}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int x^{2} - 3^{x} + \frac{1}{x} d x$

Step 4

Split the single integral into multiple integrals.

$\int x^{2} d x + \int - 3^{x} d x + \int \frac{1}{x} d x$

Step 5

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3} + C + \int - 3^{x} d x + \int \frac{1}{x} d x$

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{3} x^{3} + C - \int 3^{x} d x + \int \frac{1}{x} d x$

Step 7

The integral of $3^{x}$ with respect to $x$ is $\frac{3^{x}}{\ln (3)}$ .

$\frac{1}{3} x^{3} + C - (\frac{3^{x}}{\ln (3)} + C) + \int \frac{1}{x} d x$

Step 8

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\frac{1}{3} x^{3} + C - (\frac{3^{x}}{\ln (3)} + C) + \ln (| x |) + C$

Step 9

Simplify.

$\frac{1}{3} x^{3} - \frac{1}{\ln (3)} \cdot 3^{x} + \ln (| x |) + C$

Step 10

The answer is the antiderivative of the function $f (x) = x^{2} - 3^{x} + \frac{1}{x}$ .

$F (x) =$ $\frac{1}{3} x^{3} - \frac{1}{\ln (3)} \cdot 3^{x} + \ln (| x |) + C$